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DLM and Its Applications at CDF

This paper discusses the new formulation of the Dynamic Likelihood Method (DLM) and its applications in parton-level cross sections and likelihood, transfer functions, and the reconstruction of Mtop at CDF. It also explains the concept of DLM as a quantum process and discusses uncertainties in collider events.

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DLM and Its Applications at CDF

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  1. DLM and Its Applications at CDF March 8, 2004 @ Tsukuba K.Kondo 1. New Formulation of DLM Single path as a quantum process ★Parton level cross section and likelihood ★Transfer function Full dynamical likelihood 　２. Applications 3．　Mtop at CDF

  2. Ｄｙｎａｍｉｃａｌ　Ｌｉｋｅｌｉｈｏｏｄ　Ｍｅｔｈｏｄ量子力学的尤度法Ｄｙｎａｍｉｃａｌ　Ｌｉｋｅｌｉｈｏｏｄ　Ｍｅｔｈｏｄ量子力学的尤度法 Ｍｏｔｉｖａｔｉｏｎ Uncertainties in collider events quarks, gluons → jetsmomentum/energy measurements missing particles→MET parton-jet identification unknown initial state signal/background identification Spectroscopy: H0Collider： Lint ds Reconstruction of parton process based ontheory and observation 　　　　　　　　＜use of differential cross section＞ Traditionalds/dxmany events ＤＬＭ　 　ds/dFsingle event

  3. Ｃｏｎｃｅｐｔ　of DLMDLM : Reconstruction/Inference of Parton Process parton process theoretical experimental observed beam

  4. Differential cross section for a single event When (c1,…,cn) is given, integrate by (a,b): I(a*,b*): parton flux factor ds/dF n(f): Probability to have final phase space dFｎ(f) for a unit luminosity of the incident beam. flux(particles/cm2/sec) particles/sec luminosity(particles/cm2) particles

  5. Crosssectionandprobabilitydensityfunction (p.d.f.) Take an example of a = M. f(x;a)= p.d.f. (Axiom No.1 for p.d.f.) ( Dalitz-Goldstein,... ) This is quite reasonable for a general p.d.f., but • Nds/dx depends on the choice of x, while ds/dFn does not. • ds/dFnis a probability per unit beam luminosity. This concept is missing in the general p.d.f. f(x:M).

  6. Normalization of likelihood at parton level P(a), l0 : Prior probabilities. [ l0 ] = [L] - 2 We assume Bayes’ Postulate ( Principle of Equidistribution of Ignorance ). Quantities parameters: a variables: x, Lint and Ntot ( depend on a ). true value of a

  7. Asingle path as a quantum processQuantizedDLM For a single path, the phase volume and Jacobian- scaled transfer variables are both quantized.

  8. Phase space and observables ds/dFn(f) DFn(f) Jx D3np ds/dx* D3nx* w(y|x) D3ny* ds/dy*

  9. Resonance “sｒ” for reconstruction n – j+1 j Phase space : If parton kinematics is determined by inferring sr with propagator factor P(srｒ) , resonance r occupies a unit p.s. The cross section per unit p.s. is given by =1 where if sｒis inferred if all ci are given

  10. y dx dy x x Transfer Function defined by M.C. events • Prior TF: M.C. events with the same cuts as applied to data • Variable transformation from to : efficiency included

  11. Posterior TF x , y : indicates1-d quantity in this page TF includes efficiency.

  12. Summaryof QDLM • We assume a single path occupies a unit p.s. DFn*=1. The cross section for the path is given by • The variable space in a single path is • The likelihood for the k-th path in the i-th event is defined by • Prior TF is obtained by the MC events satisfying the event selection criteria. Posterior TF is given by • The expectation value of the event likelihood is given by • The likelihood of dynamical constant a is =1 const.

  13. Applications

  14. Correction factor (mapping function) Why? Parton-jet matching is not guaranteed. 1 parton→parton shower→fragmentation→detector→jet(s) Even if a parton and a jet are matched, topology assignment is not unique. Many solutions for unmeasured momenta

  15. Solving problems by likelihood The values of likelihood can be used for process (S vs. B) determination, right topology identification, right solution for missing momentum. By iteration, one gets better estimation of L measurement of dynamical parameter study of kinematics

  16. 戸谷・蛯名 TTbar 6 jet channel：　parton-jet idetification by Likelihood Quark level Observation level

  17. 久保 An example of L(IT) Parton ID by Moments in TTbar to l4j(1b tagged) Charged particle momentum distribution in a jet: M-. E-moments

  18. How much does ds/dF help ? TC-mass M(pT） 既知として事象再構成

  19. Mtop

  20. Top Detection • Events are energetic • Ht,SumEt • Events are central and spherical • ||< 2.0, aplanarity • Two high ET b-jets • Displaced secondary vertex • Soft lepton inside jet • High energy jets and isolated leptons • missing Et from neutrino in leptonic modes • High Et jets • Possible additional jets from gluon radiation (isr,fsr)

  21. CDF approaches to the top mass measurement History: DLM (1988) Top discovery (1994) Template, Multi-variant template DLM DZero ( Dalitz-Goldstein ) type MEAT, FLAME,MADCOW,... QDLM

  22. CDF Takes Three Paths for The Top Mass(Fermilab today 05-May-25)

  23. Why Top Mass Mt  175 GeV  Yukawa coupling  1 Special role in EWSB? Dominant parameter in radiative corrections: quadratic in mt , logarithmic in mH Mt from precision EW measurements

  24. Mtop Measurements • Combined RunI mass: • mt=178.0 ± 4.3 GeV/c2 • was: 174.3 ± 5.1 GeV/c2 • Higgs mass • Best-fit MH 113 GeV/c2 • 95% C.L. : MH < 237 GeV/c2 • Run II measurements • Systematic uncertainty largely dominated by jet energy correction: will be reduced • RunII goal is dm~2-3 GeV/c2

  25. Mtop: templates loop over j-p assignments loop over PZ for n impose kinematic constrains choose m that best fits event • Mass templete method • Reconstruct one top mass for each event • Compare Mtop distribution to simulated top templates at various masses • Minimize -ln L vs Mtop to extract mass b-tagged l+jets: Dileptons: • b-tagged l+jets w/ multivar templates: • Probability for Mt from likelihood based on MC multidimesional templates

  26. Signal/Background separationby Likelihood Distributions Parton Level CDF simulation x = Mtop y =–2ln(likelihood) background background Peak : Signal : input mass Backgrounds: lower mass Signal Signal:175GeV We expect the background makes likelihood peak down when it is multiplied to signal events. QDLM

  27. More Checks on MC vs Data More likely to be background QDLM

  28. Extracted top mass from RunII Observed :Total 22 events; electrons 12, Muons 10 Correct background-pulling 4.2 events expected. Fit : Two 2nd order polynomials for positive/negative errors. QDLM

  29. topmass ILCでの精密測定の物理 W/トップクォークの質量 ワインバーグアングル の精密測定　 →　高次補正に現れる新しい物理の精密検証 Topクォークの閾値近辺でのエネルギースキャン

  30. Conclusion • DLM is easy to use for any process. It should be the standard method in the near future. • The parton-jet, S/B and solution identifications can be improved by the likelihood itself. This is not fully demonstrated yet. • Mapping function saves you at any stage of your analysis. • It is ammusing to notice that Lagrangian Likelihood

  31. Backup slides

  32. DLM and DGM

  33. Likelihood in DLM A single path likelihood： A single event likelihood: expectation value of path likelihood Joint likelihood from multiple events：

  34. Maximum Likelihood Method (MLM) MLM in general: xi : data in i-th event, b : parameter f(x; b) : probability density function. Define Ln(b) = Pi=1n f(xi;b) , then (b0= true value of b ) MLM in DLM : ci0 :common to all paths in an event. a0:common to all events For ci0xik= c*ik, f(xik;a) = l0 ds/dc*ikPkL1ikin an event Fora0xi= y*i ,f(xi;a) = l0 <ds/dy*i >Pi L1i for multi-events parameters

  35. Tevatron: Top Production at Tevatron single top: top-antitop pairs: 85% 1.98 pb 15% 0.88 pb ~ one top event every 10 BILLION inelastic collisions

  36. Jet Energy Corrections Determine true “particle”, “parton” jet E from measured jet E • Non-linear response • Uninstrumented regions • Response to different particles • Out of cone E loss • Spectator interactions • Underlying event Jet Energy Scale (MC derived) Total Systematics

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